Doubt regarding Fourier series coefficients.












1












$begingroup$


I have some doubts while I was self studying Fourier series. I wanted to ask, there are three types of Fourier series namely; Trigonometric Fourier series, Polar Fourier series, Complex Fourier series, does complex Fourier coefficients can be found out for complex functions only? Do complex functions have non-zero Trigonometric and Polar Fourier coefficients? Do real functions have Complex Fourier coefficients? Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)? Can Complex Fourier coefficients be purely real? If any answer is 'Yes', tell me when does this happen.










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I have some doubts while I was self studying Fourier series. I wanted to ask, there are three types of Fourier series namely; Trigonometric Fourier series, Polar Fourier series, Complex Fourier series, does complex Fourier coefficients can be found out for complex functions only? Do complex functions have non-zero Trigonometric and Polar Fourier coefficients? Do real functions have Complex Fourier coefficients? Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)? Can Complex Fourier coefficients be purely real? If any answer is 'Yes', tell me when does this happen.










    share|cite|improve this question









    $endgroup$















      1












      1








      1


      1



      $begingroup$


      I have some doubts while I was self studying Fourier series. I wanted to ask, there are three types of Fourier series namely; Trigonometric Fourier series, Polar Fourier series, Complex Fourier series, does complex Fourier coefficients can be found out for complex functions only? Do complex functions have non-zero Trigonometric and Polar Fourier coefficients? Do real functions have Complex Fourier coefficients? Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)? Can Complex Fourier coefficients be purely real? If any answer is 'Yes', tell me when does this happen.










      share|cite|improve this question









      $endgroup$




      I have some doubts while I was self studying Fourier series. I wanted to ask, there are three types of Fourier series namely; Trigonometric Fourier series, Polar Fourier series, Complex Fourier series, does complex Fourier coefficients can be found out for complex functions only? Do complex functions have non-zero Trigonometric and Polar Fourier coefficients? Do real functions have Complex Fourier coefficients? Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)? Can Complex Fourier coefficients be purely real? If any answer is 'Yes', tell me when does this happen.







      fourier-analysis fourier-series






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Apr 1 '18 at 18:51







      user521346





























          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Many pages would be written for answering those questions. Briefly:



          Given a function $f$ with domain $Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.




          Does complex Fourier coefficients can be found out for complex functions only?




          Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $intlimits_{Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.




          Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?




          Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{itheta} = cos(theta) + i sin(theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.



          Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.




          Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?




          Yes.




          Can Complex Fourier coefficients be purely real?




          Yes, when the phase of that harmonic in the function is exactly zero.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2717777%2fdoubt-regarding-fourier-series-coefficients%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown
























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Many pages would be written for answering those questions. Briefly:



            Given a function $f$ with domain $Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.




            Does complex Fourier coefficients can be found out for complex functions only?




            Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $intlimits_{Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.




            Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?




            Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{itheta} = cos(theta) + i sin(theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.



            Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.




            Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?




            Yes.




            Can Complex Fourier coefficients be purely real?




            Yes, when the phase of that harmonic in the function is exactly zero.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Many pages would be written for answering those questions. Briefly:



              Given a function $f$ with domain $Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.




              Does complex Fourier coefficients can be found out for complex functions only?




              Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $intlimits_{Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.




              Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?




              Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{itheta} = cos(theta) + i sin(theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.



              Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.




              Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?




              Yes.




              Can Complex Fourier coefficients be purely real?




              Yes, when the phase of that harmonic in the function is exactly zero.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Many pages would be written for answering those questions. Briefly:



                Given a function $f$ with domain $Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.




                Does complex Fourier coefficients can be found out for complex functions only?




                Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $intlimits_{Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.




                Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?




                Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{itheta} = cos(theta) + i sin(theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.



                Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.




                Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?




                Yes.




                Can Complex Fourier coefficients be purely real?




                Yes, when the phase of that harmonic in the function is exactly zero.






                share|cite|improve this answer









                $endgroup$



                Many pages would be written for answering those questions. Briefly:



                Given a function $f$ with domain $Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.




                Does complex Fourier coefficients can be found out for complex functions only?




                Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $intlimits_{Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.




                Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?




                Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{itheta} = cos(theta) + i sin(theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.



                Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.




                Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?




                Yes.




                Can Complex Fourier coefficients be purely real?




                Yes, when the phase of that harmonic in the function is exactly zero.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 6 '18 at 22:28









                Dr PotatoDr Potato

                394




                394






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2717777%2fdoubt-regarding-fourier-series-coefficients%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Ellipse (mathématiques)

                    Quarter-circle Tiles

                    Mont Emei